User:Neil Parker/Sandbox/temp

Summary of formulae
Annual payment rate:      $$M_a=\lim_{N\to\infty}N\cdot x(N)=\frac{P_0\cdot r}{1 - e^{-rT}}$$

Future value:      $$F_v(t) = \frac{M_a}{r}(e^{rt}-1)$$

Present value:      $$P_v(t) = \frac{M_a}{r}(1 - e^{-rt})$$

Loan balance:      $$P(t) = \frac{M_a}{r}(1 - e^{-r(T-t)})$$

Loan Period:             $$T=-\frac{1}{r}\ln\left(1-\frac{P_0.r}{M_a}\right)$$

Half life of loan:      $$t_{\frac{1}{2}}=\frac{1}{r}\ln\left(\frac{1+e^{rT}}{2}\right)$$

Interest Rate:          $$r\approx\frac{2}{T}\ln{\frac{M_aT}{P_0}}$$              $$r=\frac{1}{T}\left (W(-se^{-s})+s\right )\text{ with }s=\frac{M_at}{P_0}$$